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to enter upon the study of this curve and of the various methods employed for laying it out on the ground. Those who may wish to do so will find very interesting discussions in the "Railroad Gazette," Vol. XXI, pages 1 and 93, and Vol. XII, page 639; in Engineering News," Vol. VIII, pages 214 and 246, and "Clemens's Railroad Engineer's Practice," and also a very similar curve called Froude's curve of adjustment in that well of engineering learning, "Rankine's Civil Engineering." In "Engineering News," Vol. XXIII, Mr. A. M. Wellington entered upon a discussion of the subject which, if completed, might have made this paper unnecessary, but unfortunately, he stopped short before reaching the practical application of his ideas.

It will be sufficient for our purpose to take the three following equations, which express the properties of the curve:

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In these equations, I is the abscissa and Y the ordinate of any point on the transition curve, the abscisses being measured from the PTC, or origin of the curve, along the tangent produced, and the ordinates measured from the tangent and at right angles to it; O is the ordinate of the terminal point, or PC'C' of the curve, is the length of the curve, and R is the radius, and S the shift of the cen tral curve.

We also have the following equation expressing the theoretical relations between the length of the transition curve and the elevation of the outer rail and the slope which it is necessary to give to this rail to reach this elevation:

In this equation, e

the rail rises one foot.

1 = ei

i =

(9)

elevation in feet, the distance in which Another property of the curve which we must use, is that half the length of the curve must lie on each side of what would be the PC of the circular curve, if there were no

transition curve.

If we make D

then we have —

=

the degree of curvature of the central curve,

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From equation (9), it appears that theoretically is proportional to e and i, and since in practice and theory e is made proportional to D (within certain limits), I should vary with D. The attempt to conform to this condition, is the cause of nearly all of the difficulty that is met with in the use of the cubic parabola. It may be almost entirely avoided by giving a constant value to 1.

This will have no effect upon the accuracy of the curve, for formulas (6), (7), and (8) will remain unchanged, and the curve will still be a perfect cubic parabola with varying proportions. Formula (9) is not a fundamental one, but is necessary only to adapt the curve to meet the requirements of an extreme case. Hence if we adopt a value of 1, which is the proper one for extreme values of e and i, the same value may be used for all other values of e and i, with no other effect than that the resulting transition curve will be easier than in the extreme case, which cannot be considered a fault.

For the extreme or maximum value of e, few if any railroads in this country use a greater elevation than 8 inches or foot. For the extreme or minimum value of i, Froude (quoted by Rankine with apparent approval) recommends 300 feet, or 1 inch rise in 25 feet. Substituting these values in equation (9), we have —

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The engineers of the Pittsburgh, Cincinnati, and St. Louis Railroad are said to use a value of i = 720, or 1 inch rise in 60 feet, which would make 7 = 480 feet. This would of course make an easier transition curve; but in a country where it is necessary to use curves so sharp as to require the maximum elevation (which, according to the latest practice of 1 inch elevation to each degree of curvature, would be 8° or more), it would often be impossible to find room for the two adjoining transition curves of so great length, while it would very seldom occur that a length of 200 feet could not be obtained, and the much greater simplicity of the field work makes it a very convenient length.

To find the PTC, or starting point of the transition curve, we first find the intersection of the tangents and the angle of intersection (I) in the usual way; then since half the length of the curve must lie on each side of the original PC, we find the apex distance of the PC, for the proposed D° central curve, and the angle I, either by the formula 7 R tang. I, or by the tables of apex distances, which are now to be found in all modern field

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tance of the PTC.

If in equation (8) we substitute for R its value, 5, and for its value, 200, we have

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This is the distance which the central curve is to be thrown inward toward the center. If the line has already been located in the usual manner, and it is desired to substitute a transition curve before setting the slope stakes, this can be done without the use of a transit, by simply moving the central stakes inward a distance equal to S, and for this purpose it will be sufficiently accurate to call S 0.3 D, a value very easily remembered. If a locating engineer is very much hurried, or is not perfectly familiar with the setting out of transition curves, it might be advisable to make the location in the usual manner, and leave the revision to some leisure moment. This is another point in which this is the most convenient form of transition curve.

=

By making the same substitutions in equation (7), we obtain for the ordinate, or offset from the tangent, of the P C C—

0 1.164 × D.

This is evidently equal to 48.

(13)

From equation (6) we observe that the offsets, or ordinates at intermediate points on the curve, are proportional to the cubes of the abscisses or distances of the points from the PTC measured along the tangent. Making these distances (or x in the formula) successively, 34, 31, 47, 8, 9, 37, and 1, which gives points at intervals of 25 feet, we have for the successive values of the offsets, 5120, 3120, 50, 840, 130, 180, 180, and 0; or, .00227D, .018D, .061D, .145D, .284D, .491D, 779D, and 1.164D. The transition curve may be set out by means of these offsets, and for this reason it would be well for the engineer to keep in his note-book a table of the above values of y, though the manner in which they are obtained is so plain that with a very little practice they can be easily calculated in the field if necessary.

Σ

64
2

34
312

This, however, is not a very convenient method of setting out the curve, though it may be useful for setting over the stakes of that portion of a line located in the usual manner, which lies opposite the transition curve. For this purpose the following table may be convenient :

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Column 2 contains the offsets from the tangent to the transition curve, as given above; column 3, the offsets from the tangent to the original curve calculated by the formula () n2 D, in which O is the offset, and n the distance from the PC in stations of 100 feet; and column 4, the differences between these offsets, or the distance from the original line to the transition curve. In case of necessity, this table can be easily calculated in the field.

A much simpler and better method of setting out this curve, is by deflection angles with the transit. It is true that if the distances are measured along the curve by chords, as would naturally be done, the curve ceases to be a cubic parabola, and becomes what Rankine calls Froude's curve of adjustment; but within all ordinary limits of railroad practice, the two curves are so nearly identical that they may be regarded as such without sensible error.

Supposing the transit to be setting at the P T C, and calling the deflection angle from the tangent to any point of the curve, a, we have evidently

Tang, a=

3 67R

x2 6/R

x2 D 34380 X 7

(14)

From this equation it appears that tangent a varies as 2 and D. Since within all ordinary limits of the value of D an angle varies very nearly as its tangent, we may say without material error that for all practical purposes the angle of deflection from the PT('to any point on the transition curve will vary directly as the degree of curvature of the central curve, and as the square of the distance of the point from the PTC. Hence if we have the value of a for the terminal point of the curve (or the PCC), for D = 1°, all other

values can be calculated with perfect ease in the field, or tabulated if preferred.

For this terminal point we have from equation (14), since x =

b,

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This is a very simple and easily remembered expression. We can now readily obtain the following values of a for points along the curve at intervals of 25 feet:

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=

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16

192 Do

25 D°.
1921
36 Do

192

49

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D°.
D° =

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= D°

=

11 × D.

= 20' X D.

Since the transition curve takes the place of 100 feet of tangent and 100 feet of curve on the original line, it is evident that the total central angle of the curve D°. Hence after the P C Chas been set, if we set the transit over it, take a backsight to the PTC and deflect D° · }D° = {D°, we will have the new tangent, from which the central curve can then be set out in the usual manner. If after deflecting from the PTCD° to set the PCC, we leave the vernier clamped, and take the backsight at the P C C with the vernier still reading D°, we have only to deflect till it reads D° to get the new tangent.

If, as may sometimes happen, the P T C should be inaccessible, we may set up at the original P C 100 feet farther forward on the tangent. Then since the abscissa of the PCC, when measured from this point, is half as great as when measured from the PTC, while the ordinate is the same, it is evident that the tangent of the deflection angle to the PCC or (within ordinary limits) the angle itself, will be twice as great as when taken at the PT C. Hence deflect D° from the tangent, measure 100 feet, and set the P C C. Set up at this point, backsight to the P C, and deflect D°, which will give the new tangent. The transition curve can then be set in backward. After setting out the central curve and reaching the P.CC at the other end, the second transition curve is to be set out

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